Chapter 5 exercises

Suppose $f : \{-1,1\}^n \to \{-1,1\}$ is an LTF. Show that it can be expressed as $f(x) = \mathrm{sgn}(a_0 + a_1 x_1 + \cdots a_n x_n)$ where the $a_i$’s are integers. (Hint: first obtain rational $a_i$’s by a perturbation.)

Show also that a degree-$d$ PTF has a representation in which all of the degree-$d$ polynomial’s coefficients are integers.

Let $f(x) = \mathrm{sgn}(a_0 + a_1 x_1 + \cdots a_n x_n)$ be an LTF.

Show that if $a_0 = 0$ then $\mathop{\bf E}[f] = 0$. (Hint: show that $f$ is in fact an odd function.)

Show that if $a_0 \geq 0$ then $\mathop{\bf E}[f] \geq 0$. Show that the converse need not hold.

Consider the following “correlation distillation” problem (cf. Exercise 2.49). For each $i \in [n]$ there is a number $\rho_i \in [-1,1]$ and an independent sequence of pairs of $\rho_i$-correlated bits, $(\boldsymbol{a}_1^{(1)}, \boldsymbol{b}_2^{(1)})$, $(\boldsymbol{a}_1^{(2)}, \boldsymbol{b}_2^{(2)})$, $(\boldsymbol{a}_1^{(3)}, \boldsymbol{b}_2^{(3)})$, etc. Party $A$ on Earth has access to the stream of bits $\boldsymbol{a}_1^{(1)}$, $\boldsymbol{a}_1^{(2)}$, $\boldsymbol{a}_1^{(3)}$,$\dots$. and a party $B$ on Venus has access to the stream $\boldsymbol{b}_1^{(1)}$, $\boldsymbol{b}_1^{(2)}$, $\boldsymbol{b}_1^{(3)}$, $\dots$. Neither party knows the numbers $\rho_1, \dots, \rho_n$. The goal is for $B$ to estimate these correlations. To assist in this, $A$ can send a small number of bits to $B$. A reasonable strategy is for $A$ to send $f(\boldsymbol{a}^{(1)})$, $f(\boldsymbol{a}^{(2)})$, $f(\boldsymbol{a}^{(3)})$, $\dots$ to $B$, where $f : \{-1,1\}^n \to \{-1,1\}$ is some boolean function. Using this information $B$ can try to estimate $\mathop{\bf E}[f(\boldsymbol{a}) \boldsymbol{b}_i]$ for each $i$.

Assume for now that $\rho \in [0,1)$. Show that $\boldsymbol{g}_1′, \boldsymbol{g}_2′ > 0$ if and only if $(\boldsymbol{g}_1, \boldsymbol{g}_2)$ is in a certain region $R \subseteq {\mathbb R}^2$ consisting of a quadrant plus a wedge of angle $\arcsin \rho$.